Question: Factor the following expression: $-7$ $x^2+$ $15$ $x+$ $18$
Solution: This expression is in the form ${A}x^2 + {B}x + {C}$ . You can factor it by grouping. First, find two values, $a$ and $b$ , so: $ \begin{eqnarray} {ab} &=& {A}{C} \\ {a} + {b} &=& {B} \end{eqnarray} $ In this case: $ \begin{eqnarray} {ab} &=& {(-7)}{(18)} &=& -126 \\ {a} + {b} &=& & & {15} \end{eqnarray} $ In order to find ${a}$ and ${b}$ , list out the factors of $-126$ and add them together. Remember, since $-126$ is negative, one of the factors must be negative. The factors that add up to ${15}$ will be your ${a}$ and ${b}$ When ${a}$ is ${-6}$ and ${b}$ is ${21}$ $ \begin{eqnarray} {ab} &=& ({-6})({21}) &=& -126 \\ {a} + {b} &=& {-6} + {21} &=& 15 \end{eqnarray} $ Next, rewrite the expression as ${A}x^2 + {a}x + {b}x + {C}$ $ {-7}x^2 {-6}x +{21}x +{18} $ Group the terms so that there is a common factor in each group: $ ({-7}x^2 {-6}x) + ({21}x +{18}) $ Factor out the common factors: $ x(-7x - 6) - 3(-7x - 6) $ Notice how $(-7x - 6)$ has become a common factor. Factor this out to find the answer. $(-7x - 6)(x - 3)$